When looking at the $p$-primary components of the stable homotopy ring, Hatcher says that it inherits the ring structure. I am a bit confused about this however as all the degree $0$ elements have been taken out so where exactly is the identity?

I suppose in a way I was hoping for ${}_p \pi^s_*$ to be a $\Bbb Z_p$-algebra but it seems just out of reach.

@AliCaglayan Usually people say "the p-primary part of π_*X", they mean the p-completion of π_*X (more accurately: the homotopy of the p-completion of X, but there's no difference when X is the sphere spectrum), and it is clear that that inherits the ring structure

As I said, the p-completion is better-behaved if done directly at the level of spectra, but if you're not familiar with spectra you can take the p-completion of the homotopy groups as a first approximation (they very often coincide)

I am aware that it is slightly difficult to put your hands on. But I was interested in seeing where exactly rep theory techniques break (and possibly how to fix it) Thank you for the discussion as always Denis :)

Question for the room: does anyone know a description of the set of (homotopy classes of) E_1-maps from C_2 to GL_1S? (alternatively maps of pointed spaces BC_2→BGL_1S). I think it must have a C_2 summand, but I wonder if there's anything else

@AaronMazel-Gee so what's interesting here is that the natural functor sending a differential algebra (A, d) to the DGA (A <-d- A <-d- ...) is faithful but not full? It preserves quasi-isos but doesn't reflect them. I guess that kills any hope of viewing the homotopy theory of differential algebras as a sub-theory of that of DGAs?

Oof, I kinda hate that I ended statements with question marks, but I wanted to sound less sure of myself so my poor punctuation still stands

@HarryGindi In fact, the category of trees Ω cannot be endowed with a promonoidal category structure with respect to the BV-tensor product. Indeed, such a promonoidal category would be a 6-tuple $(Ω, P, J, \lambda, \rho, \alpha)$, ...

where for any trees $Sm T$ and $U$, $P(S, T, U) = dSet(\Omega^S, \Omega^T\otimes \Omega^U)$, $J= dSet(-, \Delta^0)$, $\lambda$ and $\rho$ are isomorphisms for the tensor product of a representable presheaf $\Omega^T$ of a tree $T$ with the representable unit $\Delta^0$ and (if one compute the associated coend) $\alpha$ requires the canonical morphism of denroidal sets $(\Omega^S\otimes \Omega^T)\otimes \Omega^U \to \Omega^S\otimes (\Omega^T \otimes \Omega^U)$ to be an isomorphism

Now, we know by the paper of Heuts, Hinich and Moerdijk that $\alpha$ is not an isomorphism in general. (Note that the target of that morphism is just the dendroidal nerve of the symmetric operad $S\otimes T \otimes U$, while the source is not, in general).

@ReubenStern oh, no this is great! there's a Z-action on Ch by translation, and DiffAlg is its fixedpoints (strict/strong...maybe also homotopy?)

@ReubenStern haha it just so happens that i'm reading a book right now that uses this all the time, so i'm actually pretty used to it

"inherent vice", by thomas pynchon. it's really great. i think one might appreciate it more after reading other pynchons, but i'm sure it's good standalone too.

usually he's super-literary with all kinds of references to anything from chemistry to ancient greek philosophy to higher math to whatever... but this is, like, a pulpy detective thriller.

it's really of amusing, the diction he gives his various beach bums / stoners / deadbeats (it's set in L.A.) -- even when they're talking about acid trips and whatnot, they sometimes display these little flourishes of erudition. i get the sense that pynchon just couldn't help but make his prose beautiful.

let me use this as a springboard to strongly encourage everyone (who needs to grade things) to use Gradescope: https://gradescope.com/ it's really fantastic. it saves tons of time, the students get a much clearer picture of the rubric and where/why they lost points, you can grade at home in pajamas instead of staying at the department til 1am with your fellow graders...

it's free to use for your first two semesters. after that, you can either pay by the class, or get your department/institution to spring for a membership

Hello, Forgive the basic question, but do Grothendieck fibrations preserve limits? The fourth property [here](https://ncatlab.org/nlab/show/Grothendieck+fibration#properties) seems to imply this: limits upstairs are constructed using limits downstairs and limits in a suitable fiber, and they seem to be preserved by construction...